1. The problem involves a cone with height 12 cm and base radius 2 cm, cut and opened into a sector with angle $x=60^\circ$. We need to find the area of the shaded sector. 2. First, calculate the slant height $OM$ of the cone using the Pythagorean theorem: $$OM=\sqrt{12^2+2^2}=\sqrt{144+4}=\sqrt{148} \approx 12.17\text{ cm}$$ 3. The sector formed has radius $OM=12.17$ cm and central angle $x=60^\circ$. 4. The area of a sector is given by: $$\text{Area} = \frac{x}{360} \times \pi r^2$$ 5. Substitute $x=60$ and $r=12.17$: $$\text{Area} = \frac{60}{360} \times \pi \times (12.17)^2 = \frac{1}{6} \times \pi \times 148 = \frac{148\pi}{6} \approx 77.5$$ 6. Rounded to the nearest whole number, the area is 78 cm². 7. Among the given options, 75 cm² is the closest. **Final answer:** 75 cm²